1. Technical Field
The present invention relates in general to signal processing and in particular to a method and apparatus for spectral analysis of a signal. Still more particularly, the invention relates to a method and apparatus for real-time spectral analysis that determine two time-dependent intervals over which a signal should be analyzed to decompose it into its instantaneous mean and random components and to determine its instantaneous frequency spectrum (IFS).
2. Description of the Related Art
Spectral analysis is the decomposition of a signal into its sinusoidal components. Just as sunlight can be decomposed into a rainbow by a prism, a sampled signal can be decomposed into frequencies by an applicable algorithm. And just as a rainbow can be blended back into sunlight by passing its constituent colors through an inverted prism, a signal can be reconstructed by combining its component sinusoids.
Different signals, like chemical elements when heated to incandescence, manifest their own characteristic spectrum. A signal""s spectrum thus provides an identity or signature in the sense that no other signal has the same spectrum. From this spectrum emerge all parameters necessary to quantify the instantaneous statistical character of the signal. Phase information about the signal, however, is lost.
Hereafter, spectral analysis, unless stated otherwise, refers specifically to time-varying spectral analysis. Time-varying spectra are common in everyday life because dynamic phenomena are common in everyday life. For example, a standard musical score represents a time-varying spectrum because it indicates to a musician what notes (i.e., frequencies) should be played as time progresses. A viable time-varying spectral analysis algorithm must determine which frequencies exist in a random signal at a particular instant in time.
Conventional spectral analysis entails the decomposition of a time-varying signal into its instantaneous mean and deviations from the mean. These two components are also referred to as the systematic (or DC) component and random component, respectively. For sampled signals, the systematic and random components are independent of one another, meaning that two different signals can have the same random component but different means, and vice versa. In other words, one particular mean is not necessarily linked to one particular random part.
Presently available spectral analysis algorithms are all non-real-time and use a single user-defined averaging interval (or window) for analyzing a random signal. Specifically, conventional spectral analysis algorithms require that the user xe2x80x9cguessxe2x80x9d the requisite magnitude of the time-dependent averaging interval over which a signal is to be continually analyzed. The extent of the selected interval is a reflection of the user""s perception of how the signal properties are changing, or will change, with respect to time. (See, for example, Jones, D. L., and Parks, T. W.:1990, xe2x80x98A High Resolution Data-Adaptive Time-Frequency Representationxe2x80x99, IEEE Trans. Acous., Speech, and Sig. Proc. 38, 2127-2135; Jones, G., and Boashash, B.: 1997, xe2x80x98Generalized Instantaneous Parameters and Window Matching in the Time-Frequency Planexe2x80x99, IEEE Trans. Sig. Proc. 45, 1265-1275; and Katkovnik, V. and Stankovic, L.: 1998, xe2x80x98Instantaneous Frequency Estimation Using the Wigner Distribution with Varying and Data-Driven Window Lengthxe2x80x99, IEEE Trans. Sig. Proc. 46, 2315-2325.) Such guessing introduces biases which degrade IFS computation, because even educated guessing cannot reliably emulate changing features of random signals, in real-time or otherwise.
Often, a user employs an averaging interval of large constant extent for all portions of signal, in the belief that incorporation of a substantially large amount of information will produce a statistically valid xe2x80x9cparameterizedxe2x80x9d IFS. The parameter in question is the magnitude of the overly large constant averaging interval, which the user believes can be xe2x80x9cscaledxe2x80x9d out later. Such approaches contaminate the produced result because changes in signal properties are due to physical forces whose time-dependent features cannot be completely characterized by changes in time scales only (e.g., recall that forces have dimensions of (mass)xc3x97(acceleration)).
The use of a large constant averaging interval also contaminates the produced result whenever xe2x80x9ctoo muchxe2x80x9d information is included within the interval because signals have a natural property that defines how far back in time present characteristics are connected to earlier ones. Thus, the inclusion of too much information within the averaging interval risks the possibility that past signal behavior, which is completely unrelated to immediate behavior, is incorporated into the IFS computation. This has the same effect as two (or possibly three) pianists, for example, playing the same piece of music, but not in synchrony with one another the listener has difficulty focusing on the true melody.
Use of too much information also leads to an anomaly referred to as 1/f noise (e.g., see Keshner, M.: 1982, xe2x80x981/f Noisexe2x80x99, Proc. IEEE 70, 212-218). Use of overly long intervals also increases the likelihood that significant short-lived events (intermittency) are masked by uncharacteristically long averaging (see, e.g., Gupta, A. K.: 1996, xe2x80x98Short-Time Averaging of Turbulent Shear Flow Signalsxe2x80x99, in G. Trevixc3x1o et al. (eds.), Current Topics in Nonstationary Analysis, World-Scientific, Singapore, pp. 159-173.)
Some currently available spectral analysis algorithms also use information from the future, and in so doing assume that future behavior impacts present behavior. A common example of this approach is to define frequency information from time information according to                               F          ⁢                      xe2x80x83                    ⁢                      (                          p              ,              ω              ,              M                        )                          =                              (                          1                                                                    (                    M                    )                                    ⁢                                      xe2x80x83                                    ⁢                                      (                    SR                    )                                                                        )                    ⁢                      xe2x80x83                    ⁢                                    ∑                              n                =                p                                            p                +                M                -                1                                      ⁢                          xe2x80x83                        ⁢                          S              ⁢                              xe2x80x83                            ⁢                              (                n                )                            ⁢                              xe2x80x83                            ⁢                              exp                ⁢                                  xe2x80x83                                [                                                      -                    ⅈ                                    ⁢                                      xe2x80x83                                    ⁢                                      (                                          ω                      SR                                        )                                    ⁢                                      xe2x80x83                                    ⁢                                      (                                          n                      -                      p                                        )                                                  ]                                                                        [        1        ]            
where S(n) represents a signal in discrete form, p is an integer denoting the present, xcfx89 is frequency in radians per second, and M is a p-dependent integer assigned by the user consistent with some user-specified criterion.
For xcfx89=O, the symbol       ∑          m      =      p              p      +      M      -      1        ⁢      xe2x80x83  
denotes                               (                      1                                                            (                  M                  )                                ⁢                                  xe2x80x83                                ⁢                                  (                  SR                  )                                                              )                ⁢                  {                                    S              ⁢                              xe2x80x83                            ⁢                              (                p                )                                      +                          S              ⁢                              xe2x80x83                            ⁢                              (                                  p                  +                  1                                )                                      +                          S              ⁢                              xe2x80x83                            ⁢                              (                                  p                  +                  2                                )                                      +            …            ⁢                          xe2x80x83                        +                          S              ⁢                              xe2x80x83                            ⁢                              (                                  p                  +                  M                  -                  1                                )                                              }                                    [        2        ]            
where m takes on the integer values p,p+1,p+2, . . . ,p+Mxe2x88x921; for non-zero values of xcfx89, the symbol denotes correspondingly similar expressions. Another approach is given by                               F          ⁢                      xe2x80x83                    ⁢                      (                          p              ,              ω              ,              M                        )                          =                              (                          1                                                                    (                                                                  2                        ⁢                        M                                            +                      1                                        )                                    ⁢                                      xe2x80x83                                    ⁢                                      (                    SR                    )                                                                        )                    ⁢                      xe2x80x83                    ⁢                                    ∑                              n                =                                  p                  -                  M                                                            p                +                M                                      ⁢                          xe2x80x83                        ⁢                          S              ⁢                              xe2x80x83                            ⁢                              (                n                )                            ⁢                              xe2x80x83                            ⁢                              exp                ⁢                                  xe2x80x83                                [                                                      -                    ⅈ                                    ⁢                                      xe2x80x83                                    ⁢                                      (                                          ω                      SR                                        )                                    ⁢                                      xe2x80x83                                    ⁢                                      (                                          n                      -                      p                                        )                                                  ]                                                                        [        3        ]            
Note that both of these approaches require information beyond the present, and as such are non-causal. The formulations are also non-real-time since they require knowledge of the signal at p+1,p+2,p+3, . . . ,p+Mxe2x88x921, which denote instants in time later than p, to define a signal property at p. In a world governed by entropy and its companion irreversibility, assignations such as [1] and [3] cannot be relied on to produce even near real-time information.
Also note these approaches use the same user-defined averaging interval (M) for all components of the signal. The specific components intended here are the p-dependent DC component (mean), defined by multiplying equation [2] by
{square root over ((SR+L )/M+L )}
and the random component, defined by subtracting the p-dependent DC component from the signal. Since a signal is a time history of some random-varying physical phenomenon, there is no assurance the above user-selected M is compatible with (i.e., xe2x80x9cmatchesxe2x80x9d) the window peculiar to instantaneous properties of the signal. Accordingly, the potential for a conflict between what the user imposes and what the signal manifests is prohibitive.
Last, a popular algorithm called the Fast Fourier Transform (FFT) requires the magnitude of the analysis window to be defined by a number which is a power of 2 (i.e., 2M), where M is (again) specified by the user. As above, since a signal is a time history of some random-varying physical phenomenon, there is no assurance that the M chosen by the user matches the window peculiar to the instantaneous properties of the signal.
In view of the foregoing shortcomings in the art, the present invention provides a method and apparatus for real-time assessment of the time-dependent intervals over which a signal should analyzed to determine the signal""s IFS. From the signal and the accuracy of the sampling device, a determination is made, in real-time, of the minimum time-dependent time interval (xcex94T) over which the signal should be analyzed to separate the signal into systematic and random components. From the random component, the optimal time-dependent interval (AW) for analyzing the signal with intent to compute its IFS is determined, also in real time. Even though the properties of the signal may change over time, the properties are nonetheless taken to be constant over the extent of AW only. The use of these two real-time signal and accuracy-defined windows for signal analysis distinguishes the present invention from available algorithms, all of which are non-real-time and use the same user-defined window for analyzing both the systematic and random signal components.
For real signals, the xcex94T and AW intervals computed by the present invention are dynamic and vary randomly with time. When coupled with the periodogram version (see, e.g., Schuster, A.: 1898, xe2x80x98On the Investigation of Hidden Periodicities with Application to a Supposed 26 Day Period of Meteorological Phenomenaxe2x80x99, J. Geophys. Res., 3, 13-41) of the short-time Fourier transform (STFT), the xcex94T and AW intervals endow this STFT version of a signal with the capacity to estimate IFS. STFT is the accepted method for tracking changes with respect to time in frequency content, distribution, domain, and bandwidth, and procedures for its implementation can be found in Cohen (see, Cohen, L.: 1995, Time-Frequency Analysis, Prentice-Hall, N.J., 299). Once the instantaneous xcex94T and AW intervals and the instantaneous systematic and random components are identified, these quantities, as well as the IFS of the random component, can be continually presented to a user via a display to permit real-time visual assessment of the particular changing features of the signal.
All objects, features, and advantages of the present invention will become apparent in the following detailed written description.